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Topology/Geometry Seminar
Thursday, February 2, 2006
5:00pm,
106 McAllister
Mark Johnson
Penn State University
Factoring the Becker-Gottlieb Transfer through the Trace map
Abstract: The Becker-Gottlieb transfer of a fibration $p:E \to B$ is a stable map in the other direction, $\tau(p):Q(B_+) \to Q(E_+)$. Associated to the same fibration one also has the algebraic K-theory transfer $p^*:Q(B_+) \to A(E)$, whose target is Waldhausen's algebraic K-theory of the total space. Finally, one always has the Trace map $tr:A(E) \to Q(E_+)$ and the claim is that $tr \circ p^* \simeq \tau(p)$ for compact ANR fibrations. The proof is surprisingly clean, thanks to the axiomatic description of $\tau(p) $ for this type of fibration given by Becker and Schultz. We simply verify these axioms hold for our composite $tr \circ p^*$, although this requires us to introduce relative versions of all of the above constructions.
We meet weekly on Thursday afternoons, with the third meeting of each month in Altoona. If you are interested in giving a talk in our seminar please contact one of the coordinators for the Spring 2006 semester: Aissa Wade and Wojciech Dorabiala.
